Robust and efficient validation of the linear hexahedral element

Checking mesh validity is a mandatory step before doing any finite element analysis. If checking the validity of tetrahedra is trivial, checking the validity of hexahedral elements is far from being obvious. In this paper, a method that robustly and efficiently compute the validity of standard linear hexahedral elements is presented. This method is a significant improvement of a previous work on the validity of curvilinear elements. The new implementation is simple and computationally efficient. The key of the algorithm is still to compute Bezier coefficients of the Jacobian determinant. We show that only 20 Jacobian determinants are necessary to compute the 27 Bezier coefficients. Those 20 Jacobians can be efficiently computed by calculating the volume of 20 tetrahedra. The new implementation is able to check the validity of about 6 million hexahedra per second on one core of a personal computer. Through the paper, all the necessary information is provided that allow to easily reproduce the results, \ie write a simple code that takes the coordinates of 8 points as input and outputs the validity of the hexahedron.

[1]  Timothy C. Warburton,et al.  GPU accelerated spectral finite elements on all-hex meshes , 2016, J. Comput. Phys..

[2]  F. B. Fuller,et al.  HARMONIC MAPPINGS. , 1954, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Elaine COHEN,et al.  Rates of convergence of control polygons , 1985, Comput. Aided Geom. Des..

[4]  Steven E. Benzley,et al.  A Comparison of All Hexagonal and All Tetrahedral Finite Element Meshes for Elastic and Elasto-plastic Analysis , 2011 .

[5]  Guillaume Caumon,et al.  Indirect Unstructured Hex-dominant Mesh Generation Using Tetrahedra Recombination , 2014 .

[6]  Stephen A. Vavasis,et al.  A Bernstein-Bezier Sufficient Condition for Invertibility of Polynomial Mapping Functions , 2003, ArXiv.

[7]  Christophe Geuzaine,et al.  Geometrical validity of curvilinear finite elements , 2011, J. Comput. Phys..

[8]  Jeffrey Grandy.  Conservative Remapping and Region Overlays by Intersecting Arbitrary Polyhedra , 1999 .

[9]  Olga V. Ushakova.  Conditions of Nondegeneracy of Three-Dimensional Cells. A Formula of a Volume of Cells , 2001, SIAM J. Sci. Comput..

[10]  P. Knabner,et al.  Conditions for the invertibility of the isoparametric mapping for hexahedral finite elements , 2003 .

[11]  Christophe Geuzaine,et al.  A frontal approach to hex-dominant mesh generation , 2014, Adv. Model. Simul. Eng. Sci..

[12]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[13]  T. A. Porsching,et al.  Some results on the global inversion of bilinear and quadratic isoparametric finite element transformations , 1978 .

[14]  Bruno Lévy,et al.  Hexahedral-dominant meshing , 2017, ACM Trans. Graph..

[15]  Richard Leroy.  Certificats de positivité et minimisation polynomiale dans la base de Bernstein multivariée. (Certificates of positivity and polynomial minimization in the multivariate Bernstein basis) , 2008 .

[16]  Olga V. Ushakova.  Nondegeneracy tests for hexahedral cells , 2011 .

[17]  Kenji Shimada,et al.  Fully-automated hex-dominant mesh generation with directionality control via packing rectangular solid cells , 2003 .

[18]  Richard Leroy.  Certificates of positivity in the simplicial Bernstein basis. , 2009 .

[19]  P. M. Knupp.  On the ivertability of the isoparametric map , 1990 .

[20]  P. Knupp.  Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II—A framework for volume mesh optimization and the condition number of the Jacobian matrix , 2000 .

[21]  Jeanne Pellerin,et al.  Identifying combinations of tetrahedra into hexahedra: A vertex based strategy , 2017, Comput. Aided Des..